Harmonic Function

Harmonic functions are one of the most important functions in complex analysis, as the study of any function for singularity as well as residue, we must check the harmonic nature of the function. For any function to be Harmonic, it should satisfy the Laplacian equation, i.e., ∇²u = 0.

A harmonic function is a function that meets two criteria.

• First, it needs to be smooth, meaning it can be continuously and easily differentiated twice.
• Second, it must follow a specific rule called Laplace's equation, expressed as:

\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0 Does it refer

for a function [u(x, y)] to be harmonic, the sum of its second partial derivatives with respect to x and y must be zero.

In simple words, if any smooth function u(x, y) satisfies the equation u_xx + u_yy = 0, then this function u is a harmonic function where u_xx and u_yy represent second-order partial derivatives, with respect to x and y, respectively.

Examples of Harmonic Function

Some of the common examples of harmonic functions are:

1) Constant Function: u(x, y) = c

2) Holomorphic Function: e^x+iy

• Real Part of Holomorphic Function: e^x cos y i.e.,  Re[e^x+iy].
• Real Part of Holomorphic Function: e^x sin y i.e.,  Im[e^x+iy].

3) f(x, y) = ln(x²+ y²)

Some other examples with three variables are:

• 1/r
• x/r³
• -ln(r² - z²)
• -ln(r + z)

Where r = x² + y² + z².

What are Conjugate Harmonic Functions?

In situations where you have an analytic function ω(z) = u+iv, you can think of "v" as the conjugate harmonic function of "u" and vice versa. In other words:

If you have an analytic function ω₁(z) = u+iv, then ω₂(z) = −v+iu is also an analytic function.

In this context, u and v are considered harmonic conjugates. This means that these functions are connected specially, and when you swap the real and imaginary parts, the resulting function remains analytic.

Properties of Harmonic Functions

Some of the common properties of harmonic functions are:

• If ω(z) = u(x,y) + iv(x,y) is analytic in a region A, then both u and v are harmonic functions in A.
• When u(x,y) is harmonic in a connected region A, then u is the real part of an analytic function ω(z) = u(x,y) + iv(x,y).
• If u and v are the real and imaginary parts of an analytic function, they are considered harmonic conjugates.
• Adding two harmonic functions produces another harmonic function
• Not every pair of arbitrary harmonic functions "u" and "v" is necessarily conjugate unless u+iv forms an analytic function.

How to Identify Harmonic Function?

To identify a harmonic function, you can follow these steps.

Step 1: Understand the Basics

• Harmonic functions are smooth and have continuous second derivatives.
• They satisfy Laplace's equation: \frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0

Step 2: Examine the Function

Consider a function u(x, y), for example, u(x, y) = x² - y²

Step 3: Check Continuity and Differentiability

Ensure that u(x, y) is smooth, meaning it is continuous and has continuous first and second derivatives. In our example, u(x, y) = x² - y² is a polynomial, so it's smooth everywhere.

Step 4: Verify Laplace's Equation

Apply Laplace's equation: \(\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0\).

• Calculate \frac{\partial^2u}{\partial x^2} : (-2)
• Calculate \frac{\partial^2u}{\partial y^2} : (-2)
• Sum: 2 - 2 = 0

Since the sum is zero, the function u(x, y) = x² - y² satisfies Laplace's equation, indicating that it is a harmonic function.

Also Check

• Complex Number
• Harmonic Mean
• Harmonic Progression

Solved Examples on Harmonic Function

Example 1: Determine if the function u(x,y) = ln(x² +y²) is harmonic.

Solution:

Calculate the partial derivatives of u.

⇒ \frac{\partial^2u}{\partial x^2} = \frac{2y^2 - x^2}{(x^2 + y^2)^2}

⇒ \frac{\partial^2u}{\partial y^2} = \frac{2x^2 - y^2}{(x^2 + y^2)^2}

Sum the second partial derivatives.

⇒ \frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0

Since the sum is zero, u(x, y) = In(x² + y²) is harmonic.

Example 2: Check the harmonic nature of u(x, y) = cos(x) cosh(y).

Solution:

Compute the second partial derivatives of u.

\frac{\partial^2u}{\partial x^2} = -cos(x) cosh(y)

\frac{\partial^2u}{\partial y^2} = \os(x) cosh(y)

Sum the second partial derivatives:.

\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0

The sum is zero, indicating that u(x, y) = cos(x) cosh(y) is a harmonic function.

Practice Problems on Harmonic Functions

Problem 1: Investigate whether the function u(x, y) = x³ − 3xy² + 3x² − 3y² + 1 is harmonic.

Problem 2: Show that u(x, y) = 2x(1−y) is a harmonic function. Determine its harmonic conjugate v(x, y).

Problem 3: Prove that the function u(x, y) = e^x2−y2 cos(2xy) is harmonic. Find the harmonic conjugate v(x, y) of u, considering the ambiguity of a constant.